论文标题
不可压缩的Navier-Stokes-Cahn-Hilliard模型的全球规律性和渐近稳定性具有无与伦比的密度
Global regularity and asymptotic stabilization for the incompressible Navier-Stokes-Cahn-Hilliard model with unmatched densities
论文作者
论文摘要
我们研究了不可压缩的Navier-Stokes-Cahn-Hilliard系统的初始值问题问题，该系统具有由Abels，Garcke和Grün提出的非恒定密度。该模型在弥漫性界面理论中是针对可粘性不可压力流体的二元混合物的。在无与伦比的液体的情况下，该系统是众所周知的模型H的概括。在三个维度中，我们证明，任何全球弱解决方案（未知唯一性）都表现出规律性的传播，并稳定在平衡状态下，为$ t \ rightarrow \ rightarrow \ infty $。更准确地说，浓度函数$ ϕ $是（任意）正时Cahn-Hilliard方程的强大解决方案，而速度场$ \ Mathbf {U} $成为大型动量方程的强大解决方案。 Our analysis hinges upon the following key points: a novel global regularity result (with explicit bounds) for the Cahn-Hilliard equation with divergence-free velocity belonging only to $L^2(0,\infty; \mathbf{H}^1_{0,σ}(Ω))$, the energy dissipation of the system, the separation property for large times, a weak-strong uniqueness type结果，以及lojasiewicz-simon不平等。此外，在两个维度上，我们显示了整体系统全球强解决方案的存在和独特性。最后，我们讨论了针对双重障碍潜力的全球弱解决方案的存在。
We study an initial-boundary value problem for the incompressible Navier-Stokes-Cahn-Hilliard system with non-constant density proposed by Abels, Garcke and Grün in 2012. This model arises in the diffuse interface theory for binary mixtures of viscous incompressible fluids. This system is a generalization of the well-known model H in the case of fluids with unmatched densities. In three dimensions, we prove that any global weak solution (for which uniqueness is not known) exhibits a propagation of regularity in time and stabilizes towards an equilibrium state as $t \rightarrow \infty$. More precisely, the concentration function $ϕ$ is a strong solution of the Cahn-Hilliard equation for (arbitrary) positive times, whereas the velocity field $\mathbf{u}$ becomes a strong solution of the momentum equation for large times. Our analysis hinges upon the following key points: a novel global regularity result (with explicit bounds) for the Cahn-Hilliard equation with divergence-free velocity belonging only to $L^2(0,\infty; \mathbf{H}^1_{0,σ}(Ω))$, the energy dissipation of the system, the separation property for large times, a weak-strong uniqueness type result, and the Lojasiewicz-Simon inequality. Additionally, in two dimensions, we show the existence and uniqueness of global strong solutions for the full system. Finally, we discuss the existence of global weak solutions for the case of the double obstacle potential.
